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\bibcite{agarwal2021neural}{69}
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\bibcite{meng2022locating}{80}
\bibcite{wang2023interpretability}{81}
\bibcite{elhage2022toy}{82}
\bibcite{nanda2023progress}{83}
\bibcite{zhong2023the}{84}
\bibcite{liu2023seeing}{85}
\bibcite{elhage2022solu}{86}
\bibcite{goyal2019learning}{87}
\bibcite{ramachandran2017searching}{88}
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\bibcite{bingham2022discovering}{90}
\bibcite{bohra2020learning}{91}
\bibcite{aziznejad2019deep}{92}
\bibcite{Dubckov2011EureqaSR}{93}
\bibcite{gplearn}{94}
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\bibcite{martius2016extrapolation}{96}
\bibcite{dugan2020occamnet}{97}
\bibcite{mundhenk2021symbolic}{98}
\bibcite{yu2018deep}{99}
\bibcite{cho2024separable}{100}
\bibcite{li2020fourier}{101}
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\bibcite{maust2022fourier}{104}
\bibcite{lu2021learning}{105}
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\bibcite{kauffman2020rectangular}{107}
\bibcite{hughes2020neural}{108}
\bibcite{Craven:2020bdz}{109}
\bibcite{Craven:2022cxe}{110}
\bibcite{Ruehle:2020jrk}{111}
\bibcite{he2023machine}{112}
\bibcite{Gukov:2024aaa}{113}
\bibcite{zhang2021multiscale}{114}
\bibcite{xu2017algebraic}{115}
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